solve.\n61. amman drew a rectangle with a perimeter of 36 units. he then performed a dilation with a scale…
solve.\n61. amman drew a rectangle with a perimeter of 36 units. he then performed a dilation with a scale factor of 3. what is the perimeter in units of the resulting image? show your work.\n62. what is the x - value of the solution to the system of equations shown below?\n{y = 3x + 5\n2y = 4x + 24\n63. elizabeth wrote the number 8.45×10⁻² in standard form. which number did she write?\n64. at the café, rebecca can choose to earn $10 per hour plus a $60 starting bonus or to earn $12 per hour with no starting bonus. after how many hours of work will she earn the same amount under both payment options? write the equation to solve the problem. solve.\n65. the point (-2, -3) is rotated 180° counterclockwise about the origin. what is the y - coordinate of the resulting image?\n66. what is the slope of the line described by the data in the table below? show how you find the slope.\n|x| - 2|0|4|12|16|\n|y|5|6|8|12|14|\n67. the volume of a cone is 242.1 cubic centimeters. a cylinder has the same base and height as the cone. what is the volume in cubic centimeters of the cylinder? explain how you found the volume.\n68. at a fruit stand, rajendra can purchase three apples and one orange for the same price as five apples. the price of the orange is $0.84. what is the price in dollars of each apple? write and solve the equation.\n69. to the nearest tenth, what is the distance in units between the points (-3, 2) and (5, 6)?\n70. samantha deposited $650 into a savings account that pays 3.5% interest compounded annually. after 6 years what will be the value of her investment in dollars?\n71. how many units long is $overline{xz}$?\n72. at a farmers market the price of a basket of apples is based on the number of apples it contains. a basket that contains 12 apples costs $3.75. what is the price for a basket that contains 18 apples? show your work.
Answer
61.
Answer:
108
Explanation:
Step1: Recall dilation - perimeter rule
When a figure is dilated by a scale factor (k), the perimeter of the new figure is (k) times the perimeter of the original figure.
Step2: Identify values
The original perimeter (P_1 = 36) units and the scale - factor (k = 3).
Step3: Calculate new perimeter
(P_2=k\times P_1=3\times36 = 108) units.
62.
Answer:
2
Explanation:
Step1: Substitute (y) in the second equation
Since (y = 3x + 5), substitute (y) into (2y=4x + 24). We get (2(3x + 5)=4x+24).
Step2: Expand the left - hand side
(6x+10 = 4x + 24).
Step3: Move (x) terms to one side
(6x-4x=24 - 10).
Step4: Simplify
(2x=14), so (x = 7).
63.
Answer:
0.0845
Explanation:
Step1: Recall scientific notation rule
For (a\times10^{-n}) ((1\leqslant a<10), (n) is a positive integer), we move the decimal point (n) places to the left.
Step2: Convert (8.45\times10^{-2})
Move the decimal point 2 places to the left in 8.45, we get 0.0845.
64.
Answer:
30 hours
Explanation:
Step1: Set up equations
Let (h) be the number of hours. The first payment option is (10h + 60), and the second is (12h).
Step2: Set the two equations equal
(10h+60 = 12h).
Step3: Solve for (h)
(60=12h - 10h), (2h = 60), so (h = 30) hours.
65.
Answer:
3
Explanation:
Step1: Recall rotation rule
When a point ((x,y)) is rotated (180^{\circ}) counter - clockwise about the origin, the new point is ((-x,-y)).
Step2: Apply the rule
For the point ((-2,-3)), after rotation, the new point is ((2,3)), and the (y) - coordinate is 3.
66.
Answer:
(\frac{1}{2})
Explanation:
Step1: Recall slope formula
The slope (m=\frac{y_2 - y_1}{x_2 - x_1}). Let's take two points, say ((x_1,y_1)=(0,6)) and ((x_2,y_2)=(4,8)).
Step2: Calculate slope
(m=\frac{8 - 6}{4-0}=\frac{2}{4}=\frac{1}{2}).
67.
Answer:
726.3
Explanation:
Step1: Recall volume formulas
The volume of a cone is (V_{cone}=\frac{1}{3}\pi r^{2}h) and the volume of a cylinder is (V_{cylinder}=\pi r^{2}h).
Step2: Compare volumes
If a cylinder and a cone have the same base and height, then (V_{cylinder}=3V_{cone}).
Step3: Calculate cylinder volume
Given (V_{cone}=242.1) cubic centimeters, so (V_{cylinder}=3\times242.1 = 726.3) cubic centimeters.
68.
Answer:
(0.42)
Explanation:
Step1: Set up an equation
Let (x) be the price of each apple. The equation is (3x + 0.84=5x).
Step2: Solve for (x)
(0.84=5x - 3x), (2x = 0.84), so (x = 0.42).
69.
Answer:
8.9
Explanation:
Step1: Recall distance formula
The distance (d) between two points ((x_1,y_1)) and ((x_2,y_2)) is (d=\sqrt{(x_2 - x_1)^{2}+(y_2 - y_1)^{2}}). Here, (x_1=-3,y_1 = 2,x_2 = 5,y_2 = 6).
Step2: Substitute values
(d=\sqrt{(5+3)^{2}+(6 - 2)^{2}}=\sqrt{64 + 16}=\sqrt{80}\approx8.9).
70.
Answer:
(800.47)
Explanation:
Step1: Recall compound - interest formula
The compound - interest formula is (A=P(1 + r)^{t}), where (P) is the principal amount, (r) is the annual interest rate (as a decimal), and (t) is the number of years.
Step2: Identify values
(P = 650), (r=0.035), (t = 6).
Step3: Calculate (A)
(A=650\times(1 + 0.035)^{6}=650\times1.035^{6}\approx650\times1.23149\approx800.47).
71.
Answer:
4.5
Explanation:
Step1: Use the AA (angle - angle) similarity criterion
Since (\angle A=\angle X = 70^{\circ}) and (\angle B=\angle Y = 32^{\circ}), (\triangle ABC\sim\triangle XYZ).
Step2: Set up a proportion
The ratio of corresponding sides is equal. (\frac{AC}{XZ}=\frac{AB}{XY}). We know (AB = 2), (XY = 6), (AC = 1.5). Let (XZ=x), then (\frac{1.5}{x}=\frac{2}{6}).
Step3: Solve for (x)
(2x=1.5\times6), (2x = 9), (x = 4.5).
72.
Answer:
(5.625)
Explanation:
Step1: Find the unit price
If 12 apples cost (3.75), the unit price (u=\frac{3.75}{12}=0.3125) dollars per apple.
Step2: Calculate the price for 18 apples
The price for 18 apples is (0.3125\times18 = 5.625) dollars.